Confidence Intervals for Population Proportion: Videos & Practice Problems

In this section, we explore the construction of a confidence interval for a population proportion, denoted as \( p \). To illustrate this process, we consider a survey of 200 individuals, where 90 expressed a preference for computers from brand A over brand B. Our goal is to construct a 90% confidence interval for the true proportion of individuals who prefer brand A.

First, we calculate the sample proportion, \( \hat{p} \), which serves as our point estimate. This is done by dividing the number of successes (people preferring brand A) by the total sample size:

\( \hat{p} = \frac{x}{n} = \frac{90}{200} = 0.45 \)

Next, we need to ensure that the sampling distribution of \( \hat{p} \) is approximately normal. This is valid if both \( n\hat{p} \) and \( n(1 - \hat{p}) \) are greater than or equal to 5. In our case:

\( n\hat{p} = 200 \times 0.45 = 90 \) (successes) and \( n(1 - \hat{p}) = 200 \times 0.55 = 110 \) (failures)

Both values meet the criteria, confirming that we can proceed.

Next, we determine the critical z-value, \( z_{\alpha/2} \), corresponding to our confidence level. For a 90% confidence interval, \( \alpha = 0.10 \), thus \( \alpha/2 = 0.05 \). Using a z-table or calculator, we find:

\( z_{\alpha/2} = 1.645 \)

Now, we calculate the margin of error \( E \) using the formula:

\( E = z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \)

Substituting the values, we have:

\( E = 1.645 \times \sqrt{\frac{0.45 \times 0.55}{200}} \)

Calculating this gives:

\( E \approx 0.0579 \)

With the margin of error determined, we can construct the confidence interval by adding and subtracting \( E \) from \( \hat{p} \):

Lower bound: \( \hat{p} - E = 0.45 - 0.0579 = 0.3921 \)

Upper bound: \( \hat{p} + E = 0.45 + 0.0579 = 0.5079 \)

Thus, the 90% confidence interval for the true proportion of people who prefer brand A is \( (0.3921, 0.5079) \). This means we are 90% confident that the true proportion of individuals who prefer brand A computers lies between 39.21% and 50.79%. Understanding this interval is crucial, as it provides insight into the population's preferences based on our sample data.

In this scenario, we analyze survey data from Apple, which found that 62% of 100 sampled individuals preferred the iPhone over other smartphones. Our goal is to construct two confidence intervals: one at 99% confidence and another at 92% confidence, while understanding the implications of these intervals.

To begin, we verify that the number of successes (individuals preferring the iPhone) and failures (those who do not) are both at least 5. With 62 individuals preferring the iPhone and 38 not, this condition is satisfied for both confidence intervals.

Next, we calculate the critical value, denoted as \( z_{\alpha/2} \). For the 99% confidence interval, we find \( \alpha/2 = 0.005 \), leading to a critical z-score of \( z_{\alpha/2} = 2.576 \). The sample proportion, \( \hat{p} \), is already given as 0.62. The margin of error (E) is calculated using the formula:

\( E = z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \)

Substituting the values, we have:

\( E = 2.576 \times \sqrt{\frac{0.62 \times 0.38}{100}} \)

This results in a margin of error of approximately 0.125. Thus, the 99% confidence interval is calculated as:

\( \hat{p} - E \) to \( \hat{p} + E \), which gives us:

\( 0.62 - 0.125 = 0.495 \) and \( 0.62 + 0.125 = 0.745 \).

Therefore, the 99% confidence interval is \( (0.495, 0.745) \).

For the 92% confidence interval, we repeat the process, starting with the critical value. Here, \( \alpha/2 = 0.04 \), leading to a critical z-score of \( z_{\alpha/2} = 1.751 \). The margin of error is recalculated as:

\( E = 1.751 \times \sqrt{\frac{0.62 \times 0.38}{100}} \)

This results in a margin of error of approximately 0.085. The 92% confidence interval is then:

\( 0.62 - 0.085 = 0.535 \) and \( 0.62 + 0.085 = 0.705 \).

Thus, the 92% confidence interval is \( (0.535, 0.705) \).

Finally, we consider why these two confidence intervals differ despite being based on the same sample. The key difference lies in the confidence levels. The 99% confidence interval provides a wider range, indicating greater certainty that the true proportion of individuals preferring the iPhone lies within this interval. Conversely, the 92% confidence interval is narrower, reflecting a lower level of confidence. This illustrates the trade-off between confidence level and the precision of the estimate: higher confidence yields a broader interval, while lower confidence results in a more concise range.

In this analysis, we explore the probability of rolling snake eyes (two ones) with a pair of dice, particularly focusing on a scenario where the dice may be weighted. The initial probability of rolling snake eyes with fair dice is known to be \( \frac{1}{36} \) or approximately 2.8%. After rolling a potentially weighted pair of dice 50 times, snake eyes appeared 11 times. Our goal is to construct a 99% confidence interval for the true proportion of snake eye rolls and assess whether the dice are indeed weighted.

To begin, we verify that the number of successes (snake eyes) and failures are both greater than or equal to 5. With 11 successes and 39 failures (50 total rolls), this condition is satisfied. Next, we calculate the critical value for our confidence level. The significance level \( \alpha \) is calculated as \( 1 - 0.99 = 0.01 \), and thus \( \alpha/2 = 0.005 \). Using a z-table, we find the critical z-score \( z_{\alpha/2} \) to be 2.576.

Next, we compute the sample proportion \( \hat{p} \) using the formula \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of successes (11) and \( n \) is the total number of trials (50). This gives us \( \hat{p} = \frac{11}{50} = 0.22 \).

To find the margin of error, we use the formula:

Margin of Error = \( z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \)

Substituting the values, we have:

Margin of Error = \( 2.576 \times \sqrt{\frac{0.22 \times 0.78}{50}} \)

Calculating this yields a margin of error of approximately 0.151.

Now, we can construct the confidence interval by adding and subtracting the margin of error from the point estimate:

Confidence Interval = \( \hat{p} \pm \text{Margin of Error} \)

This results in:

Lower Bound = \( 0.22 - 0.151 = 0.069 \)

Upper Bound = \( 0.22 + 0.151 = 0.371 \)

Thus, the confidence interval is \( (0.069, 0.371) \), or 6.9% to 37.1%. This means we are 99% confident that the true proportion of snake eye rolls lies within this range.

Next, we check if the original probability of \( \frac{1}{36} \) (or 2.8%) falls within this confidence interval. Since 2.8% is less than the lower bound of 6.9%, we conclude that it does not fall within the interval.

Finally, we assess whether the dice are weighted. Given that the probability of rolling snake eyes with fair dice (2.8%) is significantly lower than our confidence interval's lower bound (6.9%), we can infer with a high degree of certainty that the dice are likely weighted.

When estimating a population proportion, constructing a confidence interval involves determining the sample size needed to achieve a desired margin of error. The margin of error (E) can be calculated using the formula:

E = z_α/2 * √(p̂(1 - p̂) / n)

In this equation, z_α/2 represents the critical z-value corresponding to the desired confidence level, p̂ is the sample proportion, and n is the sample size. To find the required sample size when the margin of error is specified, we can rearrange the formula to solve for n:

n = (z_α/2)² * p̂(1 - p̂) / E²

In a scenario where you want to estimate the proportion of students who would vote for you as student body president with 90% confidence and a margin of error of 0.02, you would follow these steps:

1. **Identify the critical z-value**: For a 90% confidence level, the critical z-value (z_α/2) is approximately 1.645.

2. **Determine p̂**: If the sample proportion (p̂) is not provided, use p̂ = 0.5. This choice maximizes the required sample size, ensuring that the margin of error is met regardless of the actual proportion.

3. **Plug values into the rearranged formula**:

n = (1.645)² * (0.5)(1 - 0.5) / (0.02)²

4. **Calculate n**: Performing the calculations yields:

n = (2.706) * (0.25) / (0.0004) = 1,691.25

5. **Round up**: Since you cannot have a fraction of a participant, round up to the nearest whole number, resulting in a required sample size of 1,692 students.

This process illustrates how to determine the minimum sample size needed to achieve a specified margin of error in estimating a population proportion, ensuring that your results are statistically valid and reliable.